Question: Moving only south and east along the line segments, how many paths are there from $A$ to $B$? [asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 9; ++i)

if (i!=4 && i !=5)

draw((2i,0)--(2i,3));
for(int j = 0; j <= 3; ++j)

draw((0,j)--(18,j));

draw((2*4,0)--(2*4,1));
draw((2*5,0)--(2*5,1));
draw((2*4,2)--(2*4,3));
draw((2*5,2)--(2*5,3));

label("$A$",(0,3),NW);
label("$B$",(18,0),E);
draw("$N$",(20,1.0)--(20,2.5),3N,EndArrow(4));
draw((19.7,1.3)--(20.3,1.3));
[/asy]
First, put the two missing segments in and count the number of paths from $A$ to $B$ on the complete grid.  Each path from $A$ to $B$ consists of a sequence of 12 steps, three of which are ``down'' and nine of which are ``right.''  There are $\binom{12}{3}=220$ ways to arrange 3 D's and 9 R's, so there are 220 paths from $A$ to $B$.

Now we will count the number of paths that go through one of the forbidden segments.  No path goes through both of them, so we may count the number of paths that go through each segment and sum the results.  Define $C$ and $D$ as shown in the figure. There are 5 ways to get from $A$ to $C$ and 6 ways to get from $D$ to $B$.  So there are $5\cdot 6=30$ ways to get from $A$ to $B$ through the first forbidden segment.  Similarly, there are 30 ways to get from $A$ to $B$ through the second forbidden segment.  So the total number of paths from $A$ to $B$ on the original grid is $220-30-30=\boxed{160}$.

[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 9; ++i)

if (i!=4 && i !=5)

draw((2i,0)--(2i,3));
for(int j = 0; j <= 3; ++j)

draw((0,j)--(18,j));

draw((2*4,0)--(2*4,1));
draw((2*5,0)--(2*5,1));
draw((2*4,2)--(2*4,3));
draw((2*5,2)--(2*5,3));

label("$A$",(0,3),NW);
label("$B$",(18,0),E);
dot("$C$",(8,2),NE);
dot("$D$",(8,1),SE);[/asy]